The Cartesian equation for a conoid with two folds is z=x2-y2x2+y2. This can be generalized to any desired number n of folds as x⁢(r,θ)=r⁢cos⁡θ, y⁢(r,θ)=r⁢sin⁡θ and z⁢(r,θ)=c⁢sin⁡(n⁢θ). Plücker’s conoid has applications in mechanical drafting.

References

1 J. Plücker, “On a new geometry of space”, Philosophical Transactions of the Royal Society of London155 (1965): 725 - 791

2 S. P. Radzevich, “A Possibility of Application of Pliicker’s Conoid for Mathematical Modeling of Contact of Two Smooth Regular Surfaces in the First Order of Tangency”, Mathematical and Computer Modelling42 (2005): 999 - 1022